Error Analysis of a Finite Difference Scheme for the Epitaxial Thin Film Model with Slope Selection with an Improved Convergence Constant

Abstract

In this paper we present an improved error analysis for a finite difference scheme for solving the 1-D epitaxial thin film model with slope selection. The unique solvability and unconditional energy stability are assured by the convex nature of the splitting scheme. A uniform-in-time $H^m$ bound of the numerical solution is acquired through Sobolev estimates at a discrete level. It is observed that a standard error estimate, based on the discrete Gronwall inequality, leads to a convergence constant of the form exp($CT\varepsilon^{-m}$), where $m$ is a positive integer, and $\varepsilon$ is the corner rounding width, which is much smaller than the domain size. To improve this error estimate, we employ a spectrum estimate for the linearized operator associated with the 1-D slope selection (SS) gradient flow. With the help of the aforementioned linearized spectrum estimate, we are able to derive a convergence analysis for the finite difference scheme, in which the convergence constant depends on $\varepsilon^{-1}$ only in a polynomial order, rather than exponential.